Solve The Equation For All Values Of $x$.$|4x + 8| + 2 = 2x$Answer Attempt 1 Out Of 2No Solution$x = \square$

Understanding Absolute Value Equations

Absolute value equations are a type of algebraic equation that involves the absolute value of a variable or expression. In this article, we will focus on solving absolute value equations of the form $|ax + b| + c = dx + e$, where $a$, $b$, $c$, $d$, and $e$ are constants.

The Given Equation

The given equation is $|4x + 8| + 2 = 2x$. Our goal is to solve this equation for all values of $x$.

Step 1: Isolate the Absolute Value Expression

To solve the equation, we need to isolate the absolute value expression on one side of the equation. We can do this by subtracting $2$ from both sides of the equation:

$|4x + 8| = 2x - 2$

Step 2: Set Up Two Separate Equations

Since the absolute value expression is equal to a linear expression, we can set up two separate equations:

$4x + 8 = 2x - 2$ or $4x + 8 = -(2x - 2)$

Step 3: Solve the First Equation

Let's solve the first equation:

$4x + 8 = 2x - 2$

Subtracting $2x$ from both sides gives:

$2x + 8 = -2$

Subtracting $8$ from both sides gives:

$2x = -10$

Dividing both sides by $2$ gives:

$x = -5$

Step 4: Solve the Second Equation

Now, let's solve the second equation:

$4x + 8 = -(2x - 2)$

Distributing the negative sign gives:

$4x + 8 = -2x + 2$

Adding $2x$ to both sides gives:

$6x + 8 = 2$

Subtracting $8$ from both sides gives:

$6x = -6$

Dividing both sides by $6$ gives:

$x = -1$

Step 5: Check the Solutions

Now that we have found two potential solutions, we need to check if they are valid. We can do this by plugging each solution back into the original equation.

For $x = -5$:

$|4(-5) + 8| + 2 = 2(-5)$

$|-20 + 8| + 2 = -10$

$|-12| + 2 = -10$

$12 + 2 = -10$

This is a contradiction, so $x = -5$ is not a valid solution.

For $x = -1$:

$|4(-1) + 8| + 2 = 2(-1)$

$|-4 + 8| + 2 = -2$

$|4| + 2 = -2$

$4 + 2 = -2$

This is also a contradiction, so $x = -1$ is not a valid solution.

Conclusion

In this article, we solved the absolute value equation $|4x + 8| + 2 = 2x$ for all values of $x$. We found two potential solutions, $x = -5$ and $x = -1$, but neither of them was valid. Therefore, the equation has no solution.

Key Takeaways

• Absolute value equations can be solved by isolating the absolute value expression and setting up two separate equations.
• When solving absolute value equations, it's essential to check the solutions to ensure they are valid.
• If a solution is not valid, it means that the equation has no solution.

Common Mistakes to Avoid

• Not isolating the absolute value expression before setting up two separate equations.
• Not checking the solutions to ensure they are valid.
• Assuming that a solution is valid without checking it.

Real-World Applications

Absolute value equations have many real-world applications, such as:

• Modeling the distance between two points on a coordinate plane.
• Representing the magnitude of a vector in physics and engineering.
• Solving problems involving absolute values in finance and economics.

Practice Problems

Try solving the following absolute value equations:

1. $|3x - 2| + 1 = 2x$
2. $|2x + 1| - 3 = x$
3. $|x - 4| + 2 = 3x$

Conclusion

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Q: What is an absolute value equation?

A: An absolute value equation is a type of algebraic equation that involves the absolute value of a variable or expression. It is typically written in the form $|ax + b| = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the absolute value expression and set up two separate equations. Then, you can solve each equation separately and check the solutions to ensure they are valid.

Q: What are the steps to solve an absolute value equation?

A: The steps to solve an absolute value equation are:

1. Isolate the absolute value expression.
2. Set up two separate equations.
3. Solve each equation separately.
4. Check the solutions to ensure they are valid.

Q: What is the difference between an absolute value equation and a linear equation?

A: An absolute value equation is a type of equation that involves the absolute value of a variable or expression, while a linear equation is a type of equation that involves a linear expression. For example, the equation $|x - 2| = 3$ is an absolute value equation, while the equation $x - 2 = 3$ is a linear equation.

Q: Can an absolute value equation have multiple solutions?

A: Yes, an absolute value equation can have multiple solutions. However, it's essential to check the solutions to ensure they are valid.

Q: What is the significance of checking the solutions in an absolute value equation?

A: Checking the solutions in an absolute value equation is crucial to ensure that the solutions are valid. If a solution is not valid, it means that the equation has no solution.

Q: Can an absolute value equation have no solution?

A: Yes, an absolute value equation can have no solution. This occurs when the solutions are not valid.

Q: What are some real-world applications of absolute value equations?

A: Absolute value equations have many real-world applications, such as:

• Modeling the distance between two points on a coordinate plane.
• Representing the magnitude of a vector in physics and engineering.
• Solving problems involving absolute values in finance and economics.

Q: How can I practice solving absolute value equations?

A: You can practice solving absolute value equations by trying the following exercises:

1. Solve the equation $|x + 2| = 5$.
2. Solve the equation $|2x - 3| = 1$.
3. Solve the equation $|x - 1| = 2$.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid when solving absolute value equations include:

• Not isolating the absolute value expression before setting up two separate equations.
• Not checking the solutions to ensure they are valid.
• Assuming that a solution is valid without checking it.

Q: Can I use technology to solve absolute value equations?

A: Yes, you can use technology to solve absolute value equations. Many graphing calculators and computer algebra systems can solve absolute value equations and provide the solutions.

Conclusion

In this article, we answered some frequently asked questions about absolute value equations. We discussed the steps to solve an absolute value equation, the difference between an absolute value equation and a linear equation, and the significance of checking the solutions. We also provided some real-world applications of absolute value equations and some common mistakes to avoid when solving them.